264 Prof. A. Anderson on Kirchhoff's 



Each of the surface integrals in the above vanishes. The 

 first is 



J J Lfidn \r j r dn \r x )\ 

 which evidently vanishes, being equal to 



The vanishing of this surface integral leads at once to the 

 expression for Green's Equivalent Stratum ; and we shall 

 see that the vanishing: of all the integrals leads to KirchhofFs 

 formula. The second integral is equivalent to 



which also clearly vanishes. The third reduces to 



J J L dn\r / r an an \r^/ r 1 dnj 



which is equal to 



Ifjt vQ-k^-^ikK^ dxdydz 



= in (— —)dxdydz = 0. 



JJJ \rir t^Vq/ 



The fourth becomes, after a little reduction, 



f ( T 2 d / 1\ 1 drf 2 d (1\ 1 drl dr Y dr n l ja 

 J^< L dn\r / r dn dnyrj r x dn dn dnj_ 



and the subject of integration of the equivalent volume 

 integral is, consequently, 



-~V 2 n+-VVo 2 -3V 2 ?', + 3V 2 ro, 

 '''o n 



which is equal to 



6 6 6 6^ 

 r i\ i\ r 



We must now show that the surface integral of the 



